In such a hands free systems, a tag carried by the user, for example in a pocket of an item of clothing, detects a magnetic field emitted by magnetic antennas arranged inside or outside the vehicle and transmits electromagnetic waves in response, for example at radio frequency, allowing the identification of the tag. To start the engine of the vehicle, the tag must be inside the vehicle but in any position and along any direction. The tag is fitted with three antennas, oriented in a pairwise orthogonal manner. Owing to the flatness of the tag, it is not possible to arrange three identical antennas in the three relevant orthogonal directions. Two ferrite core antennas are arranged, orthogonally, in the plane of the tag. Either a very short (so that its length fits into the thickness of the tag) ferrite core antenna, or an air core antenna is arranged in the direction perpendicular to the plane of the tag. In all cases, the intrinsic sensitivities of the antennas are different.
To make an isotropic tag, that is to say one which interacts with the vehicle while exhibiting the same sensitivity whatever the orientation of the tag, it is known to reduce, at the level of the antennas, the sensitivity of the most sensitive antennas so as to align them with the sensitivity of the least sensitive antenna. These arrangements have the drawback that, during use of the tag outside the vehicle, for which use isotropy is not required, all the antennas exhibit low sensitivity, equal to the lowest intrinsic sensitivity of the three relevant antennas. The range of the tag is then needlessly reduced on account of the implementation of the arrangements indicated hereinabove.